Focused Gaussian beam going through a plane
You can create such a beam using a Volumetric Emission shader and some maths nodes.

The Gaussian distribution is as follows :

 

Create a mesh (eg, a cube) to act as the domain for the volumetrics. The Object coordinates will be used, passed through a Mapping node to allow it to be easily manipulated and scaled.

To to create a vertical beam you can calculate the square of the distance as 'x^2 + y ^ 2'. This can be used directly as the 'x^2' in the above equation.

All that remains is to scale the beam based on the square of the Z-coordinate.

This produces the following material :



You can adjust the Mapping node to vary the scaling along each of the X, Y, Z axis, adjust the 'Add' node to affect how tightly the beam is focused and adjust the Multiply node to affect the overall brightness of the beam.



Blend file attached 

EDIT : Just noticed the edit to add variation in intensity towards the focus. This can be easily achieved by driving the final Multiply for the intensity from the Z coordinate. eg, something like the following to focus the beam to a bright spot on the plane :